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Isoperimetric bounds for higher eigenvalue ratios for the n-dimensional fixed membrane problem

Published online by Cambridge University Press:  14 November 2011

Mark S. Ashbaugh  and
Rafael D. Benguria
Mark S. Ashbaugh
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A. e-mail: mathmsa@mizzou1.missouri.edu
Rafael D. Benguria
Affiliation:
Facultad de Física, P. Universidad Católica de Chile, Avda. Vicuña Mackenna 4860, Casilla 306, Santiago 22, Chile e-mail: rbenguri@lascar.puc.cl
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Synopsis

We give several results which extend our recent proof of the Payne-Pólya–Weinberger conjecture to ratios of higher eigenvalues. In particular, we show that for a bounded domain Ω⊂ℝn the eigenvalues of its Dirichlet Laplacian obey where λm denotes the mth eigenvalue and jp,k denotes the kth positive zero of the Bessel function Jp(x). Certain extensions of this result are given, the most general being the bound where k≧2 and l(m) denotes the number of nodal domains of an mth eigenfunction. Our results imply certain further conjectures of Payne, Pólya, and Weinberger concerning λ3/λ2 and λ4/λ3. In addition, we find a resonably good bound on λ4/λ1. We also briefly discuss extensions to Schrödinger operators and other elliptic eigenvalue problems.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 6 , 1993 , pp. 977 - 985
Copyright
Copyright © Royal Society of Edinburgh 1993

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